bstract arxiv:1706.09072v1 [stat.ap] 27 jun 2017 · influencenetworksininternationalrelations...

INFLUENCE NETWORKS IN INTERNATIONAL RELATIONSSHAHRYAR MINHAS, PETER D. HOFF, AND MICHAEL D. WARD

ABSTRACT. Measuring influence and determining what drives it are persistent questions

in political science and in network analysis more generally. Herein we focus on the do-

main of international relations. Our major substantive question is: How can we deter-

mine what characteristics make an actor influential? To address the topic of influence,

we build on a multilinear tensor regression framework (MLTR) that captures influence

relationships using a tensor generalization of a vector autoregression model. Influence

relationships in that approach are captured in a pair of n×nmatrices and provide mea-surements of how the network actions of one actor may influence the future actions

of another. A limitation of the MLTR and earlier latent space approaches is that there

are no direct mechanisms through which to explain why a certain actor is more or less

influential than others. Our new framework, social influence regression, provides a wayto statistically model the influence of one actor on another as a function of character-

istics of the actors. Thus we can move beyond just estimating that an actor influences

another to understanding why. To highlight the utility of this approach, we apply it to

studying monthly-level conflictual events between countries as measured through the

Integrated Crisis Early Warning System (ICEWS) event data project.

Date: June 29, 2017.Shahryar Minhas and Michael D. Ward acknowledge support from National Science Foundation

(NSF) Award 1259266 and Peter D. Hoff acknowledges support from NSF Award 1505136.

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Influence Networks June 29, 2017

MOTIVATIONThe concepts of power and influence are gold-standard building blocks in analyses of world

politics. Well-known debates on these blocks in early works include Von Clausewitz (1832); Haas

(1953); Fucks (1965); Keohane and Nye (1977); Baldwin (1978); Rummel (1979); Waltz (1979). In

the 1980s and 1990s, scholars also struggled with the meaning of power and influence in world

affairs (Doran and Parsons, 1980; Keohane, 1989; Gowa and Mansfield, 1993). This continues

to the present as scholars look forward as well as backward into history (Kadera, 2001; Barnett

and Duvall, 2004; Nexon, 2009). Despite the nearly ubiquitous use of the concept of power,

and its derivative concept influence, there is no agreed upon way to assess this feature of worldpolitics. Indeed not only is there no agreement on how to measure this concept, there is little

agreement on exactly what it means conceptually. Is power based on military capabilities?

On, diplomatic prowess? The notions of power and influence imply relations among political

actors such as countries, and as such they may be best thought of a relational characteristics

rather than material ones. Herein we address a way of examining one type of influence within

a specific network framework. The ability of one actor in world politics to influence another is

intimately tied to other actors in the world. Sometimes it is quite overt as when the US calls for

China to influence North Korea. Other times these linkages and dependencies may be more

diffuse and spread out, as when a coalition of 31 countries influence Iraq to withdraw its forces

from Kuwait. We develop a network approach to show how these kinds of dependencies can

be studied.

Relational data defines connections between pairs of actors – also known as dyads. These

connections can take the shape of simple undirected, binary relations observed for a snapshot

in time to complex types of directed and weighted relations that are observed longitudinally.

The study of these types of relations is almost ubiquitous in scholarly work. In genetics, re-

searchers have defined actors as proteins and links as the bonds between them (Wu et al., 2009;

Welch, Bansal and Hunter, 2011; Livi et al., 2016). Whereas in the social sciences, researchers

have applied network analysis to the study of terrorist networks, legislators in Congress, and

the diffusion of civil wars (de Bie et al., 2017; Cho and Fowler, 2010; Metternich, Minhas and

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Ward, 2015). The study of these types of data is made more interesting by the possibility that

the relations observed do not arise or evolve independent of one another. Observations in

relational data may be simultaneously dependent on all other observations due to the social

ties and pathways that give shape to the global structure in which actors are embedded. This

dependence is why many study relational data not as a set of independent dyadic observa-

tions, but as a network in which the link between a pair of actors influences and is influenced

by other dyads. Characterizing the manner in which observations are interdependent and then

using those interdependencies to examine the emergence and evolution of a network is a prin-

cipal focus of social network analysis.

A popular approach framework through which to characterize interdependencies are latent

variablemodels, such as the latent class (Snijders and Nowicki, 1997), distance (Hoff, Raftery and

Handcock, 2002), and factor models (Hoff, 2005). Each of these model broader patterns—such

as homophily and stochastic equivalence—as a function of node-specific latent variables. These

approaches are effective at characterizing influence patterns that emerge in the network, but

they are only able to explain those patterns through endogenous explanations. For example,

actors that cluster together in the Euclidean space estimated from latent distance models or

that are assigned to similar blocks by latent class models are assumed to possess some set of

similar characteristics based on dependence patterns in the network. Yet, these approaches

leave unanswered the question of what those characteristics are?

To address this broader question, we build on the bilinear network autoregression model in-

troduced by Hoff (2015) and Minhas, Hoff and Ward (2016). At its core, this approach is a vector

autoregression model extended to handle relational data. Within this approach dependen-

cies between observations are captured by a pair of n × n matrices that measure sender- and

receiver-level influence patterns. Themodel takes the following form: yij,t =∑aii′bjj′xi′j′,t−1+

eij . The term ai,i′ captures how previous actions of i′affect those of i and bj,j′ shows how ac-

tions towards target j are influenced by prior actions toward j′. To characterize influence pat-

terns via a set of exogenous attributes we rewrite the influence parameters so that they depend

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on covariates. This enables us to reduce the bilinear model into a rank one regression model:

yij,t = α′Xij,t−1β + eij that we refer to as the social influence regression (SIR) model.1

The simplification of the bilinear autoregression model allows us to incorporate actor and

dyad-level covariate information into determining influence patterns within the network. We

apply this approach to data from the Integrated Crisis Early Warning System (ICEWS) event data

project. The ICEWS event data are constructed by applying natural language processing and

graph theory techniques (Boschee, Natarjan and Weischedel, 2013) to a corpus of about 30

million media reports from about 275 local and global news sources in or translated to English.

Each media report is coded in accordance with an ontology of events that is derived from the

Conflict and Median Event Observation (CAMEO) scheme (Schrodt, Gerner and Yilmaz, 2009).

Our focus for this project is centered around modeling monthly level material conflict events

as tracked by ICEWS. With the SIR model we can estimate the extent to which actors within

the material conflict network influence one another, and, for example, explore whether or not

characteristics such as a pair of countries being allied is related to the influence of one on

another. Finally, we show that this network-based approach to understanding the evolution of

the material conflict network has substantially better out-of-sample performance than extant

approaches employed in the literature.

METHODSBilinear network autoregressionmodel. Many studies examine the flows or linkages amongactors, such as whether two countries are in a conflict with one another. Data from such studies

can be thought of relational data which is often represented in the form of a matrix as shown

in Figure 1. This matrix n × n where n denotes the number of actors in the network. The off-

diagonals of represent the interaction that took place between two actors, so yij represents

an interaction that took place between actors i and j. In the case of undirected data, this

1Rank regression (Izenman, 1975) is an approach to regression for data that do not conform to the

normal Gaussian assumptions. In contrast to standard approaches, a Rank Regression imposes no real

distributional assumptions on the underlying data. In particular rank regression bases its calculations

on information about the ranks of the dependent variables. This also makes the resultant models less

sensitive to outliers in the data, in the same way that the median is less influenced by outliers than the

mean.

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may simply be an indicator that i and j are allied to one another. For directed data, the rows

designate the senders of a particular action and the columns the receiver, so the yij entry

would represent an action sent from i → j. The diagonals are typically undefined indicating

that actors do not interact with themselves.

Yt =

i . . . j . . . k

i NA . . . yji . . . yik...

.... . .

.... . .

...

j yji . . . NA . . . yjk...

.... . .

.... . .

...

k yki . . . ykj . . . NA

Figure 1. Matrix represen-tation of a dyadic, rela-

tional measure for one time

point.

Figure 2. Array representa-tion of a longitudinal dyadic

measure. Darker shading

indicates later time periods.

Figure 1 represents the interactions that take place between actors for a snapshot in time.

In many fields, such as international relations, a single cross-section of data is insufficient, and

we observe a time series of interactions between countries. Extending network approaches to

studying longitudinal networks has become a topic of recent attention (Snijders, 2014; Krivit-

sky and Handcock, 2014; Sewell and Chen, 2015; Schein et al., 2015). To represent longitudinal

network structures, we begin by binding adjacency matrices into an array (see Figure 2). Specif-

ically, let Y = {Yt : t = 1, . . . , T} be a time series of sociomatrices of relational data where T

represents the number of time points, and the dimensions of this object are n × n × T . Esti-

mating models on structures such as these is the focus of the bilinear autoregression model.

The basis of this approach is a first-order vector autoregression model in which we regress the

network at one point in time on its lag. The parameters that captures the relationship between

them are a pair of matrices that capture sender and receiver dependence patterns for each

pair.

More concretely, a generalized bilinear autoregression model for Y is given:4


E[yi,j,t] = g(µi,j,t)

{µi,j,t} =Mt = AXtB>

{µi,j,t} = a>i Xtbj

where xi,j,t is a function of yi,j,t, such as x̃i,j,t ∼ log(yi,j,t−1 + 1).

In the next section we explore an example with count data. Therein Y is a time series of

matrices of counts of events between actors. Accordingly, we model yi,j,t ∼ Poisson(eµi,j,t),

where x̃i,j,t = log(yi,j,t−1 + 1). The basis of this framework is still a generalized bilinear model

so this approach is readily extendable to other distribution types. A and B are n× n “influence

parameters.” The value of aii′ captures how predictive the actions of country i′at time t − 1

are of the actions of country i at time t, while the value of bjj′ captures how predictive the

actions directed at country j′ at time t− 1 are of the actions directed towards country j at time

t. For example, consider a bilinear autoregression model on conflict that includes the United

Kingdom (GBR) and the United States of America (USA). If we estimate that aGBR,USA is greater

than zero, this implies that countries the USA initiated/continued a conflict with in period t− 1

are likely to also face a conflict from GBR in period t. Thus, GBR’s future actions are influenced

by the USA, or, put more concretely, actions of the USA are predictive of GBR’s.

Social influence regression. The SIR model explains the influence in terms of covariates andallows us to determine what makes an actor influential. Particularly, to determine the charac-

teristics of i or i′ that are related to the influence aii′ , we consider a linear regression model

for aii′ and bjj′ , given by aii′ = α>wii′ and bjj′ = β>wjj′ , where wii′ is a vector of nodal and

dyadic covariates specific to pair ii′ that we are using to estimate influence. The application we

present in the following section has time-varying covariates, which this model is able to account

for through time varying influence parameters: aii′t = α>wii′t and bjj′t = β>wjj′t.

The network autoregression model can be expressed as:

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µi,j,t =∑i′j′

aii′txi′j′tbjj′t =∑i′j′

α>wii′txi′j′tw>jj′tβ

= α>

∑i′j′

xi′j′twii′tw>jj′t

β = α>X̃ijtβ

Typically, yi,j,t also has covariates. For example, we might want to condition estimation of

the parameters on a lagged version of the dependent variable, yi,j,t−1, a measure of reciprocity,

yj,i,t−1, and other exogenous variables. In the case of estimating a model on material conflict

between a pair of countries, this might include other exogenous aspects such as the geograph-

ical distance between a pair of countries. These additional exogeneous parameters can be

accommodated with a model of the form:

µi,j,t = θ>zi,j,t + α>X̃ijtβ,

where zi,j,t represents the design array incorporating parameters that may have a direct

effect on the dependent variable. The model presented here is a type of low-rank matrix

regression: it regresses the outcome yij,t on the matrix Xij,t. An unconstrained (linear) re-

gression would be expressed as µij,t = θ>zij,t + 〈C,Xij,t〉, where C is an arbitrary p × p ma-

trix of regression coefficients to be estimated. In contrast, the regression specified above re-

stricts C to be rank one, that is, expressible as C = αβ>. This follows from the identity that

〈αβ>, Xij,t〉 = α>Xij,tβ. Low rank matrix regression models have been considered by Li, Kim

and Altman (2010) and Zhou, Li and Zhu (2013).

Estimation. To estimate the parameters, {θ, α, β} we employ an iterative process because themodel is bilinear. Specifically, for a fixed β the model is linear in (θ, α). For fixed α the model is

linear in (θ, β). Hence:6


µi,j,t = (θ> α>)

zi,j,t

X̃ijtβ

= (θ> β>)

zi,j,t

X̃>ijtα

Maximum likelihood estimate can be obtained with an iterative block coordinate descent

method for estimation of θ, α and β. Given initial values of β, iterate the following until conver-

gence:

(1) Find the conditional maximum likelihood estimate (MLE) of (θ, α) given β using iterative

weighted least squares (IWLS);

(2) Find the conditional MLE of (θ, β) given α using IWLS.

Using this approach the problem of finding the conditional MLEs turns into a sequence of low

dimensional generalized linear model (GLM) optimizations.2For example, let n be the number

of nodes, p be the length of each wii′ vector, and q be the length of each zij,t vector. Then step

one from above can be implemented as follows:

(a) Let x̃ij,t be the vector of length p+ q obtained by concatenating zij,t andXij,tβ.

(b) Construct the matrix X̃ having n× (n− 1)× T rows and p+ q columns, where each row is

equal to x̃ij,t for some (directed) pair i, j at time t.

(c) Let y be the vector of length n × (n − 1) × T consisting of the entries of Y = {Y1, . . . , YT },

ordered to correspond to the rows of X̃ .

(d) Obtain the MLEs for the Poisson regression of y on X̃ . From the regression coefficients,

extract the (conditional) estimates of θ and α.

Step 2 of the iterative algorithm works similarly, by replacingXij,tβ in item (a) withX>ij,tα.

Inference. Approximate standard errors and confidence intervals for the parameters can beobtained from the derivatives of the log-likelihood function at the MLE. This claim, however,

2Implementing this type of model is relatively straightforward using base functions such as glm in

statistical software such as R, but the code to run these type of models is available in a package that willbe hosted on CRAN and/or the corresponding author’s github.

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comes with a caveat: The multiplicative parameters α and β are not identifiable, as the term

α>Xβ is the same as (α/c)>X(cβ) for any scalar c. Meaningful derivative-based standard er-

rors need to be derived from an identifiable parameterization of the model. An identifiable

parameterization may be obtained by placing a scale restriction on α or β, or fixing one ele-

ment of either. The identifiable parameterization employed herein restricts the first element of

α to be one.

Log-likelihood derivatives of the identifiable parameters may be obtained by calculating the

derivatives for the unconstrained, non-identifiable parameterization, and then using the chain

rule to obtain the derivatives for the constrained, identifiable case. Let H be the matrix of

second derivatives of the log-likelihood at the MLE ψ̂ of ψ = (θ, α, β) (using an identifiable

parameterization). An estimate of the variance of ψ̂ is given by H−1, and standard errors for ψ̂

are given by the square roots of the diagonal elements fofH−1. The asymptotic validity of these

standard errors relies upon the assumption that the underlying model is correct. Alternatively,

model-robust standard errors can be obtained using a Sandwich variance estimate,

V̂ar(ψ̂) = H−1SH−1

where the matrix S is given by

∑i,j,t

(yi,j,t − µi,j,t)L̇ij,tL̇>ij,t,

with L̇ij,t denoting the derivative of the log-likelihood corresponding to the single observa-

tion yij,t. In the application that follows we utilize model-robust standard errors.

Figure 3 provides a visual summary of this model. The array in the far left represents the

network being modeled, the design array in green represents explanatory variables used to

directly model linkages between dyads, and the θ vector includes the estimates of the effect

those variables have on the network. To capture dependence patterns, a logged and lagged

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version of dependent variable are included, along with a design array containing a set of in-

fluence covariates,W ; α and β are vectors that capture parameter estimates for the effects of

those influence covariates. A benefit of this framework is that once estimated, linear combi-

nations of the influence regression parameters permit visualization the resulting sender and

receiver dependence patterns in the network.

Figure 3. Visual summary of social influence regression model.

EMPIRICAL APPLICATIONICEWS Material Conflict. Over the past few years, a number of projects have arisen seekingto create large data sets of dyadic events through the automatic extraction of information from

on-line news archives. This has made it empirically easier to study interactions among coun-

tries, as well as among actors such as NGOs within countries.

The two most well-known developments include the ICEWS event data project (Boschee

et al., 2015a) and the Phoenix pipeline (OEDA, 2016). At present, the field of event data isevolving, but ICEWS remains the gold standard. For the purposes of this project we focus

on utilizing the ICEWS database which also extends back farther in time. ICEWS draws from

over 300 different international and national focused publishers (Boschee et al., 2015b). TheICEWS event data are based on a continuous monitoring of over 250 news sources and other

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open source material covering 177 countries worldwide. ICEWS consists of several compo-

nents, including a database of over 38 million multilingual news stories going back to 1990

and present to last week. The ICEWS data along with extensive documentation have been

made publicly available (with a one year embargo) on dataverse.org (Lautenschlager, Shell-

man and Ward, 2015; Boschee et al., 2015a). To classify news stories into socio-political topics,ICEWS relies on a augmented and expanded version of the CAMEO coding scheme (Schrodt,

Gerner and Yilmaz, 2009). The dictionaries, aggregations, ground truth data, and actor and

verb dictionaries are publicly available with a one year lag at the ICEWS data repository https:

//dataverse.harvard.edu/dataverse/icews. In addition, the event coder has been made

available publicly by the Office of the Director of National Intelligence.3This event coder, known

as ACCENT, searches for the following information: a sender, a receiver, an action type, and a

time stamp. The set of action types covered include activities between dyads such as “Occupy

territory”, “Use conventional force”, and “Impose embargo, boycott, or sanctions”. Then, the

ontology provides rules through which the parsed story is coded. An example of a coded news

story fitting this last category is:

“President Bill Clinton has imposed sanctions on the Taliban religious faction thatcontrols Afghanistan for its support of suspected terrorist Osama bin Laden, the WhiteHouse said Tuesday.”

In this example, the actor designated as sending the action is the United States and the

actor receiving it is Afghanistan. Dyadic measurements such as these are available for 249

countries, and the dataset is updated regularly. Currently, data up until March 2016 has been

made publicly available on the ICEWS dataverse.

Our sample for this analysis focuses on monthly level interactions between countries in the

international system from 2005 to 2012.4To measure conflict from this database we focus on

what is often referred to as the “material conflict” variable. This variable is taken from the “quad

variable” framework developed by Duval and Thompson (1980). Schrodt and Yonamine (2013)

3Details at http://bit.ly/2nS4nBU.4The ICEWS data extends to 2016 but we end at 2012 due to temporal coverage constraints among

other covariates that we have incorporated into the model.

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dataverse.org

https://dataverse.harvard.edu/dataverse/icews

https://dataverse.harvard.edu/dataverse/icews

http://bit.ly/2nS4nBU


defines the type of events that get drawn into this category as those involving, “Physical acts of

a conflictual nature, including armed attacks, destruction of property, assassination, etc.”.

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Figure 4. Network depiction of ICEWS Material Conflict events for January 2005(top) and December 2012 (bottom).

Figure 4 visualizes the material conflict variable as a network, specifically, we provide snap-

shots of events between dyads along this relational dimension in January 2005 and December

2012. The size of the nodes correspond to how active countries are in the network, and each

node is colored by its geographic position. An edge between two nodes designates that at least

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one material conflict event has taken place between that dyad, and arrows indicate the sender

and receiver. Thicker edges indicate a greater count of material conflict events between a dyad.

In both snapshots, the United States is highly involved in conflict events occurring in the

system both in 2005 and 2012. Additionally, other major powers such as Russia and the Great

Britain are also frequently involved. Some notable changes are visible in the network. While

in 2005 Iraq was highly involved in material conflict events by 2012 Syria became more active.

Last, there is a significant amount of clustering by geography in this network. Conflict involving

Latin American countries is relatively infrequent but when it does occur, it seems to primarily

involve countries within the region.

Parameters with direct effect. What variables have a direct effect on the level of conflictbetween countries? There are a number of the standard explanations provided in the conflict

literature. Inertia and reprocity top the list. Conflict in period t is affected by what occurred

previously in period t − 1. This is autoregressive dependence. The expectation is that a dyad

engaged in conflict in the previous period is more likely to be engaged in conflict in the next.

A lagged reciprocity parameter embodies the common argument that if country j receives

conflict from i in period t, that in period t + 1 j may retaliate by sending conflict to i. The

argument that reciprocity is likely to occur in conflict networks is certainly not novel, and has

its roots in well known theories involving cooperation and conflict between states (Richardson,

1960; Choucri and North, 1972; Rajmaira and Ward, 1990; Goldstein, 1992).

A number of exogenous explanations have often been used to explain conflicts between

dyads. One of the most common relates to the role of geography. Apart from conflict involving

major powers, conflict between countries that are geographically proximate is typical (Bremer,

1992; Diehl and Goertz, 2000; Carter and Goemans, 2011). Figure 4 provides some evidence for

the tendency of conflict to occur between countries within the same region. The minimum,

logged distance between the dyads operationalizes this covariate.5

One of the most well developed arguments linking conflict between dyads to domestic in-

stitutions involves the idea of the democratic peace. The specific vein of this argument that

5Minimum distance estimation was conducted using the CShapes package (Weidmann, Kuse and

Gleditsch, 2010).

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has found the most support is the idea that democracies are unlikely to go to war with one

another (Small and Singer, 1976; Maoz and Abdolali, 1989; Russett and Oneal, 2001). Arguments

for why democracies may have more peaceful relations between themselves range from how

they share certain norms that make them less likely to engage in conflict to others hypothe-

sizing that democratic leaders are better able to demonstrate resolve thus reducing conflict

resulting from incomplete information (Maoz and Russett, 1993; Fearon, 1995). To operational-

ize this argument, we construct a binary indicator that is one when both countries in the dyad

are democratic.6

We also control for whether or not a pair of countries are allied to one another using data

from the Correlates of War (Gibler and Sarkees, 2004).7Typically, one would expect that states

allied to one another are less likely to engage in conflict. Another common control in the con-

flict literature is the level of trade between a pair of countries. We estimate trade flows between

countries using the International Monetary Fund (IMF) Direction of Trade Statistics (Interna-

tional Monetary Fund, 2012). Incorporating the level of trade between countries speaks to a

long debate on the role that economic interdependencies may play in mitigating the risk of

conflict between states (Barbieri, 1996; Gartzke, Li and Boehmer, 2001).8

The last set of measures we use to predict dyadic conflict are derived from another ICEWS

quad variable. Verbal cooperation counts the occurrence of statements expressing a desire

to cooperate from one country to another.9We include a lagged and reciprocal version of

this variable to our specification. This monthly level measure of cooperation between states

provides us with a thermometer measure of the relations between states that is measured at a

low level of temporal aggregation.

6We define a country as democratic if its polity score is greater than or equal to seven according to

the Polity IV project (Marshall and Jaggers, 2002).7We consider a pair of countries allied to one another if they share amutual defense treaty, neutrality

pact, or entente.

8The extant literature has employed a variety of parameterizations to test this hypothesis. At times,

a measure of trade dependence is calculated and at others just a simple measure of the trade flows

between a pair of countries. We show results for the latter parameterization but results are consistent

if we utilize a measure of trade dependence.9An example of a verbal cooperation event sent from Turkey to Portugal is the following: “Portugal

will support Turkey’s efforts to become a full member of the European Community, Portuguese President MarioSoares said on Tuesday.”

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Parameters defining influence patterns. The novel feature of the SIR model is its ability toexplain influence patterns as a function of an underlying regression model estimated jointly

with the parameters directly modeling yij through the iterative procedure described in the pre-

vious section.

Thus using the SIR model we can answer the following types of questions:

• Are actions directed at the one country at time t − 1 predictive of the actions directed

towards another at time t?

• What characteristics can explain why the actions of the one country at time t − 1 are

predictive of the actions of another country at time t?

The first covariate added to the influence specification, is simply a control for the distance

between countries.10A negative effect for the distance parameter in the case of sender influ-

ence would indicate that countries are likely to send conflictual actions to the same countries

that their neighbors are sending conflictual actions too. In the case of receiver influence, a neg-

ative effect would indicate that countries are likely to be targeted by the same set of countries

that their neighbors are receiving conflictual interactions from.

An interesting argument that has received continuing attention in the political science liter-

ature is the role that alliances play in either mitigating or increasing the level of conflict in the

international system. A number of scholars have argued that in the case of a conflict, a coun-

try’s allies will join in to honor their commitments thus increasing the risks for a multiparty

interstate conflict (Snyder, 1984; Leeds, 2003; Vasquez and Rundlett, 2016). We would find evi-

dence for this argument if the ally parameter in the case of sender influence was positive, as

that would indicate that countries are more likely to initiate or increase the level of conflict with

countries that their allies are in conflict with.

Last, we include measures for the level of trade and verbal cooperation between countries.

Interpretations for how the effects of these covariates may play out follows a similar framework

to what has been described above. Table 1 summarizes each of the covariates used to estimate

the social influence regression on the material conflict variable from ICEWS.

10This is operationalized similarly as above using data from CShapes.

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Zijt

Material Conflictij,t−1 Allyij,t−1Material Conflictji,t−1 Log(Trade)ij,t−1Distanceij,t−1 Verbal Cooperationij,t−1Joint Democracyij,t−1 Verbal Cooperationji,t−1

Wijt

Distanceij,t−1Allyij,t−1Log(Trade)ij,t−1Verbal Cooperationij,t−1

Table 1. Model specification summary for social influence regression. Top rowshows covariates used to estimate direct effects and bottom sender and receiver

influence.

Parameter Estimates. Figure 5 depicts the parameter estimates using a set of coefficient

plots.11On the left, we summarize the estimates of the direct effect parameters. As expected,

greater levels of conflict between a dyad in the last period are associated with greater levels of

conflict in the present. This speaks to a finding common in the conflict literature regarding the

persistence of conflicts between dyads (Brandt et al., 2000). We also find evidence that coun-

tries retaliate to conflict aggressively, though this effect is imprecisely measured. In terms of

our exogenous parameters, the level of conflict between a dyad is negatively associated with

the distance between them, a finding that aligns well with the extant literature.

Additionally, as is typical in the extant literature we find that jointly democratic dyads are

unlikely to engage in conflict with one another. Surprisingly, however, the level of trade be-

tween countries is positively associated with the level of conflict. The divergence of this finding

with some of the extant literature may be a result of a variety of factors, such as our use of

a measure of conflict that has much greater variance than the militarized interstate disputes

measurement from the Correlates of War dataset. Or, it may be a consequence of having the

network dependencies more fully specified for the first time.

11Convergence diagnostics are presented in Figure A1 of the Appendix.

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VerbalCooperationji,t-1

VerbalCooperationij,t-1

Log(Trade)ij,t-1

Allyij,t-1

JointDemocracyij,t-1

Distanceij,t-1

MaterialConflict ji,t-1

MaterialConflict ij,t-1

-0.50 -0.25 0.00 0.25

SenderInfluenceEstimates


Log(Trade)ij,t-1

Allyij,t-1

Distanceij,t-1

-2 -1 0 1

ReceiverInfluenceEstimates


Log(Trade)ij,t-1

Allyij,t-1

Distanceij,t-1

-0.008 -0.004 0.000

VerbalCooperationji,t-1


Log(Trade)ij,t-1

Allyij,t-1

JointDemocracyij,t-1

Distanceij,t-1

MaterialConflict ji,t-1

MaterialConflict ij,t-1

-0.50 -0.25 0.00 0.25

SenderInfluenceEstimates


Log(Trade)ij,t-1

Allyij,t-1

Distanceij,t-1

-2 -1 0 1

ReceiverInfluenceEstimates


Log(Trade)ij,t-1

Allyij,t-1

Distanceij,t-1

-0.008 -0.004 0.000

Figure 5. Left-most plot shows results for the direct effect parameters and thetop-right plot represents results for the sender influence, and bottom-right re-

ceiver influence parameters. Points in each of the plots represents the average

effect for the parameter and the width the 90 and 95% confidence intervals.

Dark shades of blue and red indicate that the parameter is significant at a 95%

confidence interval and lighter shades a 90% confidence interval. Parameters

that are not significant are shaded in grey.

The right-most plots focuses what determines sender (top) and receiver (bottom) influence

patterns. Interestingly, the alliance sender influence parameter has a positive effect, indicating

that countries tend to initiate greater levels of conflict with countries that their allies were fight-

ing in the previous period. This finding is in line with arguments in the extant literature about

16


the role that alliance relationships may play in leading to more conflict in the international sys-

tem (Siverson and King, 1980; Leeds, 2005).

Additionally, countries are likely to send conflict to those with whom their verbal cooperation

partners are initiating or increasing conflict with. This finding is interesting as it highlights that

countries making cooperative statements regarding a particular country i, actually go beyond

those statements in later periods to supporting i by initiating conflict with those that i was

in conflict with. Trade flows, on the other hand, are associated with having a negative effect,

implying that countries are not likely, and in fact somewhat unlikely, to follow their trading

partners into conflict.

Receiver influence patterns are similarly determined. Trade flows and verbal cooperation

have similar effects, though the interpretation here for trade is that countries are unlikely to

be targeted by those that target their trading partners. Interestingly, the distance effect on

the receiver influence side is more precisely measured, implying that geographically proximate

countries are more likely to receive conflict from a similar set of countries.

Visualizing Dependence Patterns. Based on the sender and receiver influence parameterestimates, Figure 6 provides a visual summary of the type of dependence patterns that are

implied in the context of the material conflict model estimated in the previous section.

The linear combination of our influence parameter estimates (α), and the design array con-

taining sender influence variables (wijt) are used to visualize the sender dependence patterns

between a pair of countries (aijt): aijt = α>wijt. The resulting sender and receiver dependence

pattern are shown in Figure 6 for June 2007.12For the visualization on the left [right], edges

between countries indicate that greater likelihood to send [receive] conflictual events to [from]

the same countries. Countries are colored by their relative geographic position and node size

corresponds to the number of influence relationships the country shares.

Since these dependence patterns are estimated directly from the model results that are pre-

sented in Figure 5, the patterns implied by that model are manifest in these visualizations. One

12A lengthier table of visualizations for additional time periods is shown in Figure A2 of the Appendix.

17


of the more notable findings from the sender influence model is the role that alliance relation-

ships play, and this effect is striking. For example, the USA shares sender influence ties with

a number of Western European countries such as Germany and the United Kingdom, the USA

also is more likely to send conflict to actors that Australia, South Korea and Japan have engaged

in material conflict with, and many of these countries are likely to do the same.

JUNE 2007SENDER DEPENDENCE PATTERNS

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Figure 6. Network visualization of influence patterns as estimated by the socialinfluence regression model for June 2007. Nodes are colored by their relative

geographic position and are sized by the number of influence relationships that

they receive and send.

18


A predictor of receiver influence patterns is the distance between countries. Countries are

more likely to be targeted by the same set of countries as their neighbors. This pattern mani-

fests itself in the right-most visualization in Figure 6, where we find clumps of countries, such

as Iraq, Lebanon, and Jordan, clustering together.

Performance Comparison. A common and important argument for employing a networkbased approach is that it aids in better accounting for the data generating process underly-

ing relational data structures.

Thus, in this case, the network approach should actually better predict conflict in an out-of-

sample test. To put the performance of this model in context, we compare it to a standard

GLM that does not account for dependence patterns in the network, but is similarly parameter-

ized. Additionally, given the recent interest in machine learning methods as tools for prediction

within the social sciences we compare the performance against a generalized boosted model

(GBM).

Boosting methods have become a popular approach in the machine learning to ensemble

over decision tree models in a sequential manner. At each iteration, a new model is trained

with respect to the error of the ensemble at that point. Friedman (2001) greatly extended the

learning procedure underlying boosting algorithms, by modifying the approach to choose new

models at every iteration so that they would bemaximally correlated with the negative gradient

of some loss function relevant to the ensemble. In the case of a squared-error loss function,

this would correspond to sequentially fitting the residuals. We use a generalized version of

this model developed by Ridgeway (2012) that extends this framework to the estimation of a

variety of distribution types—in our case, a Poisson regression model. In general, these types

of models have been shown to give substantial predictive advantage over alternative methods,

such as GLM, and should provide a useful point of comparison.13

To compare these approaches we first utilize a cross-validation procedure. This involves first

randomly dividing T time points in our relational array into k = 10 sets and within each set we

set randomly exclude five time slices from our material conflict array. We then run our models

13The R gbm package on CRAN implements this estimator (Ridgeway, 2012).

19


and predict the five missing slices from the estimated parameters. Proper scoring rules are

used to compare predictions. Scoring rules evaluate forecasts through the assignment of a

numerical score based on the predictive distribution and on the actual value of the dependent

variable. Czado, Gneiting and Held (2009) discuss a number of such rules that can be used

for count data: Brier, Dawid-Sebastiani, Logarithmic, and Spherical scores.14For each of these

rules, lower values on the metric indicate better performance.

Dawid-Sebastiani RMSE

Logarithmic Brier Spherical

20 40 60 0.10 0.15 0.20

0.6 0.7 0.8 -1.75 -1.70 -1.65 -1.60 -20 -15 -10 -5

Social InfluenceModel

Generalized BoostedModel

Generalized LinearModel

Figure 7. Performance comparison based on randomly excluding time slicesfrom the material conflict array. Colors designate the different models, and the

average score across the 10-fold cross validation is designated by a circle and the

range by a horizontal line.

Figure 7 illustrates differences in the performance between the social influence model, GLM,

and GBM across the scoring rules mentioned above and a more standard metric, the RMSE. In

the case of each of these metrics we find GLM performs the worst and that the social influence

model performs the best.

14Details are provided in the Appendix.

20


Recently, conflict scholars have become interested in generating forecasts from their models

(Pevehouse and Goldstein, 1999; de Mesquita, 2009; Brandt, Freeman and Schrodt, 2014; Hegre,

Høyland and Nygård, 2015; Ward, 2016). and to assess the performance of their model instead

of taking a cross-validation approach they often just predict out some number of years. We

perform such an exercise as well by dividing up our sample into a training and test set, where

the test set corresponds to the last x periods in the data that we have available. We vary x from

two to five. For instance, when x = 5 we are leaving the last five years of data for validation.

Results for this analysis are shown in Figure 8 and there again we find that the social influence

model has better out of sample predictive performance than the alternatives we test here.

Dawid-Sebastiani RMSE

Logarithmic Brier Spherical

Y T-2:T

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Social InfluenceModel

Generalized BoostedModel

Generalized LinearModel

Figure 8. Performance comparison based on randomly excluding the last twoto five periods of the material conflict array. Colors and shapes designate the

different models, and the score when excluding x number of periods is shown.

21


CONCLUSIONBoth measuring influence and determining the drivers of it are topics of perennial interest in

network analysis and political science. Within political science, determining whether and how

actors within relational systems influence one another is a topic of particular interest in regards

to deriving measures surrounding the relative “power” of states. Though much past work has

drawn upon explanations based on power to assess world events, extant operationalizations

of this concept can be improved.

The standard approach of assessing the relative power of a pair of states is derived by calcu-

lating the ratio of their material capabilities (e.g., Slantchev 2004; Reed et al. 2008; Butler and

Gates 2009; Gartzke and Weisiger 2014). The availability of quantitative data on material char-

acteristics of states (e.g., population, iron and steel production, military size) was to influence

greatly the scholarship in this area. Currently, many scholars continue to rely on indices such

as the Correlates of War’s Composite Index of National Capabilities (CINC) as a way of assessing

power (Singer, Bremer and Stuckey, 1972). For the most part, this has pushed scholarly consid-

eration of power into a capabilities direction, rather than a direction in which power was seen

as relational.

These approaches have implicitly assumed that power is material and fungible. If China hasmore capabilities than India, it has more power. If India and Japan together have more capabili-

ties than China, then they havemore power. Yet, relying on these types of measures ignores the

nuances of regional as well as global interactions, and disregards the contexts in which states

interact. Further, the narrow interpretation of power characterized by purely detracts from a

more relevant question regarding relational data, namely, how do the actions of actors within a

network influence the actions of others. Through using the approach we have introduced here

scholars can continue to test theories regarding the role that alliances or trade flows may play

in influencing states, but can move towards doing so within a network context.

The work that we have done is also relevant for the networks literature. Discussions around

measurements of influence often begin and end with the use of various centrality measures.

Yet, centrality measures just provide a representation of how “important” a node is within a

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network, and do not detail how a pair of nodes might be influencing the actions of one an-

other. Further centrality measures for the most part are just descriptive tools. Of course one

can shift towards using alternative approaches such as latent variable models, but these ap-

proaches cast little light on exogenous attributes that might be shaping how actors within a

system influence one another in a longitudinal context. The approach that we introduce here

is an extension of earlier work involving the bilinear autoregression model, and we have now

simplified it into a rank one regression model. This approach allows us to estimate the role

that nodal and dyadic attributes may play in how dyads influence one another, and because

this approach is estimated within a GLM framework it is readily extendable to a variety of other

settings.

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APPENDIXVisualization of convergence for direct (blue), sender influence (green), and receiver influence

(red) parameters.

Verbal Cooperationji,t-1

Log(Trade)ij,t-1 Verbal Cooperationij,t-1

Joint Democracyij,t-1 Allyij,t-1

Material Conflictji,t-1 Distanceij,t-1

Intercept Material Conflictij,t-1

2 4 6 8

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0.00085

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Distanceij,t-1 Allyij,t-1

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0.0050Verbal Cooperationji,t-1


Joint Democracyij,t-1 Allyij,t-1

Material Conflictji,t-1 Distanceij,t-1

Intercept Material Conflictij,t-1

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Figure A1. Convergence diagnostics for the social influence regression modelon material conflict.

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Convergence.Influence Dynamics. Visualization of influence effects for select time points from dynamicsocial influence regression model.

SENDER INFLUENCE SPACE:

FEBRUARY 2005 SEPTEMBER 2006 JUNE 2007

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Figure A2. Influence relationships25


Scoring Rules for Count Data. Scoring rules are penalties s(y, P ) introduced with P beingthe predictive distribution and y the observed value. The goal of researchers interested in

prediction is to minimize the expectation of these scores, which is typically calculated by taking

the average:

S = 1n

∑ni=1 s(yi, Pi),

where yi refers to the ithobserved count and Pi the i

thpredictive distribution. A set of

proper scoring rules as defined by Czado, Gneiting and Held (2009) are shown in the list below.

For each of these rules, f(y) denotes the predictive probability mass function. µ̂ and σ̂ refer to

the mean and standard deviation of the predictive distribution.

• Dawid-Sebastiani score: s(y, P ) = (y−µ̂σ̂ )2 + 2× log(σ̂)

• Logarithmic score: s(y, P ) = −log(f(y))

• Brier score: s(y, P ) = −2f(y) +∑

k f2(k)

• Spherical score: s(y, P ) = f(y√∑k f

2(k)

26


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SHAHRYAR MINHAS: DEPARTMENT OF POLITICAL SCIENCE

Current address: Duke UniversityE-mail address, Corresponding author: [email protected]

PETER D. HOFF: DEPARTMENT OF STATISTICS

Current address: Duke UniversityE-mail address: [email protected]

MICHAEL D. WARD: DEPARTMENT OF POLITICAL SCIENCE

Current address: Duke UniversityE-mail address: [email protected]

33

bstract arxiv:1706.09072v1 [stat.ap] 27 jun 2017 · influencenetworksininternationalrelations...

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